Download Network Information Flow

Network Information Flow
Nikhil Bhargava (2004MCS2650)
Under the guidance of
Prof. S.N Maheshwari
(Dept. of Computer Science and Engineering)
IIT, Delhi
Overview of the Presentation
•
•
•
•
•
•
Introduction to the Problem.
Previous work done.
Analysis of Butterfly network.
Results and Conclusions.
Future research focus.
References.
Introduction to the problem
• Aim is to improve the throughput in
single/multiple source network multicast
scenario.
• Throughput will be low since conventional
switching of information will time share the links
(details to follow).
• Could be improved by doing coding (i.e. XOR
operations on the incoming bit streams) at
nodes or vertices of the network graph.
• Problem is to find the maximum information flow
for a point/multi-point to multi-point multicast
network.
Introduction to the problem
(contd.)
• Each node in a conventional network
functions as a switch which:
– Either replicates information received
– Forwards info. from input link to output links
• Network coding is used to boost throughput.
• In this approach, each node receives
information from all the input links, encodes it
(i.e. combines it by the XOR operations) and
sends information to the output link.
Previous work
• Ahlswede et. al., in their seminal work [1] have
introduced a new class of problems of finding
maximum admissible coding rate region for a
generic communication multicast network.
• They gave a upper bound based on max-flow
min-cut theorem for the information flow that
could be achieved by network coding.
• Note that in network coding, the flow is not
preserved.
• Max-flow min-cut theorem in graph theory
literature is for information flow preserving
networks.
Difference between Fluid flow
and information flow
Network switching
Network coding
Previous work (contd.)
• Li, Yeung, and Cai [2] showed that the
multicast capacity can be achieved by
linear network coding for acyclic networks
i.e. networks having a graph with no
cycles.
• Yeung, Li, Cai, and Zhang in [4] have
done an extensive survey on the theory of
network coding.
Motivation for the problem
• Liang [6] have given a game theoretic approach
to solve single source network switching for a
given communication network.
• He computed network coding gain from maxflow min-cut bound.
• Based upon certain conditions on link capacities,
a game matrix is constructed and solved to give
the maximum flow using network switching.
• The network considered is not general and there
are no known ways to provide analytical
solutions to maximum flow
– It has to be done on a case by case basis.
Motivation for the problem
• There is no formal way to compute maximum
flow for a generic network.
• The game theory approach needs
computation for each network on a case by
case basis.
• Need an alternative to determine network
switching gain (i.e. maximum information flow
by switching) for a single source multicast
network.
• Need to understand the problem of multisource multi-sink network coding.
Network Coding
• Intermediate nodes transmit packets that
are functions of the received packets.
• Mostly linear functions are used.
– Known result that linear functions are enough
to achieve the max-flow min-cut bound for both
cyclic and acyclic networks.
• Can make the network robust to link
failures.
• Peer-to-peer multicast file sharing network
• Wireless Networks, sensor, adhoc, mobile.
Work done
• Started analyzing butterfly network [1] to
find its switching gap.
• Switching gap for a network is defined
as the ratio of maximum achievable
information rate using network coding
(NC) to that of network switching (NS).
• Enumerated min-cuts and calculated
NC using max-flow min-cut theorem
• Enumerated multicast routes for each
sink and created a game matrix for it.
• Solved the matrix to get maximum
achievable information rate due to
network switching.
• Calculated switching gap and analyzed
it for different cases of link capacities.
• Extended the network by taking its dual
and triple version and then analyzed
each of them.
Singular Symmetric Butterfly
network
Min-cuts for one sink in Singular
Symmetric Butterfly network
• The sub graph for sink t1 has following
7 s-t cuts
1. {(s,a), (s,b)} = 2w1
2. {(s,a), (b,c)} = w1+w2
3. {(s,a), (a,c), (d,t1)} = 2w2+w3
4. {(a, t1), (a,c), (b,c)} = w1+w2+w5
5. {(s,a), (a,c), (c,d)} = w1+w2+w4
6. {(a, t1), (c,d)} = w3+w4
7. {(a, t1), (d, t1)} = w3+w5
• The sub graph for sink t2 has following 7
s-t cuts
1.{(s,a), (s,b)} = 2w1
2.{(s,b), (a,c)} = w1+w2
3.{(b, t2), (a,c), (b,c)} = 2w2+w3
4.{(d, t2), (s,b), (b,c)} = w1+w2+w5
5.{(s,b), (b,c), (c,d)} = w1+w2+w4
6.{(b, t2), (c,d)} = w3+w4
7.{(b, t2), (d, t2)} = w3+w5
Max. Information flow due to
network coding
•
Assuming following conditions on link
capacities
1. w1<w2
2. w1<w3
3. w4<w5
Maximum information flow due to
network coding is
w1+min(w1, w4)
Max. Information flow due to
network switching
• Rows denote edges and columns denote multicast
routes.
• It has nine edges and 7 multicast routes.
• Edges (s,a); (s,a) and (c,d) are dominating edges
(a,t1),(a,c); (b,t2),(b,c) and (d, t1), (d, t2) respectively.
• Maximum information flow due to network switching
is
(2w1+min(2w1, w4))/2
• Switching gap comes out to be
2(w1+min(w1, w4))/(2w1+min(2w1, w4))
• In case all edge capacities are equal, switching gap
comes out to be 4/3.
Dual Symmetric Butterfly network
Triple Symmetric Butterfly
network
Results
• I have analyzed special cases of Ahlswede’s
butterfly network to find switching gap.
• Used game theory to calculate the maximum
information flow using switching case.
• Based upon observations for singular, dual and
triple version of the above network, gave an
intuitive result for generic class of above network
• Game theory principles fails to solve the matrix
for generic case.
Conclusions
• For the chosen class of networks
– The max-flow min-cut bound remains
constant.
– Information flow achieved by network
switching decreases as one increases the
size of the network.
• Thus overall switching gap increases
– Network coding becomes more useful for
large graphs.
Future Focus
• Find network switching gain using mincut trees for single source and later
multi-source networks (open problem).
• Find network coding gain for multisource multicast network (open
problem).
• Find optimum switching strategy for a
specific class of graphs.
References
[1] Ahlswede, N. Cai, S.-Y. Li, and R. W. Yeung, “Network
information flow,” IEEE Trans. Inform. Theory, vol. 46,
no. 4, pp. 12041216, July 2000.
[2] S.-Y. Li, R. W. Yeung, and N. Cai, “Linear network
coding,” IEEE Trans. Inform. Theory, vol. IT-49, no. 2,
pp. 371381, Feb. 2003.
[3] Christina Fragouli, JeanYves Le Boudec, Jorg Widmer,
“Network Coding: An Instant Primer,” LCA-REPORT2005-010.
[4] R. W. Yeung, S.-Y. Li, N. Cai, and Z.Zhang,“Theory of
network coding,” submitted to Foundations and Trends
in Commun. and Inform. Theory, preprint, 2005.
References
[5] C.K Ngai and R. W. Yeung, “Network switching gap
of combination networks,” in 2004 IEEE Inform.
Theory Workshop, 24-29, Oct 2004, pp.283-287.
[6] Xue-Bin Liang, “On the Switching Gap of AhlswedeCai-Li-Yeung’s Single-Source Multicast Network,”
2006 IEEE Int. Symp. Inform. Theory, Seattle,
Washington, USA, July 2006.
[7] Xue-Bin Liang, “Matrix Games in the Multicast
Networks: Maximum Information Flows With Network
Switching,” IEEE Trans. Inform. Theory, vol. 52, no.
6, June 2006.
[8] G. Owen, Game Theory, 3rd edition, San Diego:
Academic Press, 1995.